Two-Step Estimation and Inference with Possibly Many Included Covariates

TL;DR

This paper develops a bias correction method using the jackknife for two-step estimators with many covariates, improving inference accuracy in microeconomics applications.

Contribution

It introduces a jackknife-based bias correction and bootstrap inference method for two-step estimators with many covariates, addressing bias and inference issues.

Findings

Jackknife bias correction effectively reduces bias.

Bootstrap inference provides valid post-correction confidence intervals.

Method performs well in simulations and applied microeconomics settings.

Abstract

We study the implications of including many covariates in a first-step estimate entering a two-step estimation procedure. We find that a first order bias emerges when the number of \textit{included} covariates is "large" relative to the square-root of sample size, rendering standard inference procedures invalid. We show that the jackknife is able to estimate this "many covariates" bias consistently, thereby delivering a new automatic bias-corrected two-step point estimator. The jackknife also consistently estimates the standard error of the original two-step point estimator. For inference, we develop a valid post-bias-correction bootstrap approximation that accounts for the additional variability introduced by the jackknife bias-correction. We find that the jackknife bias-corrected point estimator and the bootstrap post-bias-correction inference perform excellent in simulations, offering important improvements over conventional two-step point estimators and inference procedures, which are not robust to including many covariates. We apply our results to an array of distinct treatment effect, policy evaluation, and other applied microeconomics settings. In particular, we discuss production function and marginal treatment effect estimation in detail.